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To obtain rates of redox processes from Marcus theory two factors are required, .... L is the distance from a point in the cell to its nearest image; ...
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J. Phys. Chem. B 2007, 111, 10800-10806

Can Marcus Theory Be Applied to Redox Processes in Ionic Liquids? A Comparative Simulation Study of Dimethylimidazolium Liquids and Acetonitrile R. M. Lynden-Bell† Department of Chemistry, UniVersity Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom, and Atomistic Simulation Centre, School of Maths and Physics, Queen’s UniVersity, Belfast BT7 1NN, United Kingdom ReceiVed: June 4, 2007; In Final Form: July 13, 2007

Simulations of a model system of charged spherical ions in the ionic liquids dimethylimidazolium chloride, [dmim][Cl], dimethylimidazolium hexafluorophosphate, [dmim][PF6], and the polar liquid acetonitrile, MeCN, are used to investigate the applicability of Marcus theory to electrochemical half-cell redox processes in these liquids. The free energy curves for solvent fluctuations are found to be approximately parabolic and the Marcus solvent reorganization free energies and activation free energies are determined for six possible redox processes in each solvent. The similarities between the different types of solvent are striking and are attributed to the essentially long-range nature of the relevant interactions and the effectiveness of the screening of the ion potential. Nevertheless, molecular effects are seen in the variation of solvent screening potential with distances up to 2 nm.

1. Introduction Room temperature ionic liquids are proving to be useful solvents for electrochemistry. Although some examples are viscous, they have high electrical conductivities and a large electrochemical window.1 From the point of view of a physical chemist they are a very different type of liquid to normal electrolytes, which are solutions of salts in polar solvents such as water or dimethylformamide. They are more similar to hightemperature molten salts, but have large organic cations which lower the freezing point to around room temperature. It is common to interpret redox properties in electrolyte solutions in terms of Marcus theory,2,3 which assumes that the polarization response of the solvent is a linear function of the field. It is relevant to ask whether this is still true for ionic liquids. Further developments of Marcus theory have used a continuum model for the solvent with different zero and highfrequency values of the relative permittivity (dielectric constant). If Marcus theory is applicable to redox processes in ionic liquids then one question is what is the relevant value of the zero frequency permittivity. To obtain rates of redox processes from Marcus theory two factors are required, the activation free energy of the process and the rate at which the system attains the transition state. In classical Marcus theory the latter is inversely proportional to the longitudinal relaxation time, which is related to molecular reorientation. Here the question is what quantity replaces this in an ionic liquid. Molecular simulation has proved useful in investigating redox processes in solution in polar solvents. Warshel and coworkers4,5 showed how the free energy curves describing solvent fluctuations could be studied using molecular simulations. This was followed by several groups6-9 for different systems. Much of the work is concerned with the simulation of charge transfer between explicit sites on molecules, for example, as the result of photoexcitation of dyes. Electrochemical redox processes † Address correspondence ot this author at University Chemical Laboratory. E-mail: [email protected].

differ in a number of ways. It has been shown that half reactions may be treated separately and that the relative free energy of the oxidized and reduced states is tuned by the external electrical potential.10 Recently Sprik and co-workers10-13 have used ab initio methods to study this problem. Ionic liquids are readily studied by classical molecular simulation.14-16 Kim and co-workers,17-19 Kobrak and coworker,20,21 and others have addressed the problem of the solvent response to photoexcitation of dyes in ionic liquids. The simulation of redox processes in the electrochemical context is a new application that we address here. There are two main differences between the electrochemical case and the photoexcitation case. The first is that the two-half reactions may be treated separately and the second is that the relative free energy of the oxidized and reduced states can be tuned by the external electrical potential.10 By simulating a simple model system of spherical ions with different charges in three solvents, two ionic liquids dimethylimidazolium hexafluorophosphate ([dmim][PF6]) and dimethylimidazolium chloride ([dmim][Cl]) and one polar liquid (acetonitrile, MeCN), we are able to compare polar and ionic liquids and to determine the extent to which Marcus theory applies to these liquids. We determined the values of the solvent reorganization free energies and activation free energies for six different redox processes in each solvent. One surprising result is the similarity of the values of these quantities in the ionic and polar liquids and the reason for this is discussed in section 4 in terms of continuum models. However, the molecular nature of the solvent is shown to be important and to affect the screening of the dissolved ion up to distances of several nanometers. The mechanism of screening is also different in the two types of liquid; in polar liquids it is primarily the result of molecular reorientation, while in ionic liquids it is primarily the result of ion translation. Although this leads to differences in the prefactor in the kinetics, detailed investigation of this quantity will be addressed in future work. The current study

10.1021/jp074298s CCC: $37.00 © 2007 American Chemical Society Published on Web 08/22/2007

Redox Processes in Ionic Liquids

J. Phys. Chem. B, Vol. 111, No. 36, 2007 10801

∫ exp[-β(qeφs + ESR IS + ESS)]δ(X -

concentrates on free energies and screening which are equilibrium properties. A preliminary account of this work has appeared.22

A(q,X) ) -kBT ln

2. The Model and Calculations

which can be rewritten

2.1. The Model System. The model system comprises a single ion with charge +qe dissolved in one of the three solvents [dmim][PF6], [dmim][Cl], or MeCN. The ion, I, is a similar size to a chloride ion and has charges q ) +3, +2, ..., -2, -3 in different simulations. As a result six possible redox processes can be studied

I+q f I+(q+1) + e-

(2)

The value of µ can be tuned by the external potential applied to the electrochemical cell. In classical simulations it is most convenient to use the solvent contribution VIPsolv as the order parameter X, and this is what is used in this paper. In our model systems, which were simulated under conditions of constant volume, the Helmholtz free energy A(q) of a solution containing an ion with charge q is given by

A(q) ) -kT ln Λ-3N

A(q,X) ) qX + Asolv(X)

∫ exp[-β(qeφs + ESR IS + ESS)] dτ

(3)

where ESR IS is the non-Coulombic parts of the ion-solvent interaction and ESS contains all the solvent-solvent interactions. φs is the electrical potential due to the solvent at the solute ion. The integral is over the whole 3N dimensional configuration space of solvent molecules and ion. We note that the solvent contribution to the vertical ionization energy is VIPsolv ) eφs, and this is the quantity we shall use as the solvent order parameter X. We can define a function A(q,X), the constrained or Landau free energy of a system with solute charge q and solvent order parameter X, by

(5)

where Asolv(X) is a universal solvent function that is independent of the charge on the ion and is given by

Asolv(X) ) -kBT ln

∫ exp[-β(ESR IS + ESS)]δ(X - eφ) dτ

(6)

(1)

where the values of q range from -3 to +2. This is in the same spirit as the study of Hartnig and Koper,7 who studied the solvent reorganization free energies and other properties of spheres of variable charge in water. Marcus2 introduced many of the ideas used to describe solvent effects on redox reactions. He proposed that such reactions can only occur when the fluctuations in the solvent are favorable. If one defines an order parameter X describing these fluctuations, then the probabilities of fluctuations in both oxidized and reduced states are described by the free energy curves Fox(X) and Fred(X). The point at which these curves cross gives the activation free energy, while the free energy differences at the minima of either curve Xox or Xred give the solvent reorganization free energies λox and λred, respectively. Marcus pointed out that if the solvent fluctuations were quadratic, then the free energy curves in each state would be parabolic with the same width, λox ) λred, and that the activation free energy would be equal to λ/4. Finally he developed expressions for these quantities in a polar solvent using a continuum model. The instantaneous energy gap between the reduced and oxidized form is a useful quantity to take as the order parameter,4 and for an electrochemical half reaction this is just equal to the vertical ionization energy.10 In such an electrochemical half reaction the vertical ionization energy can be written as the sum of the gas-phase ionization energy IEgas, a solvent contribution VIPsolv, and the electrochemical potential of the electron µ

VIP ) IEgas + VIPsolv + µ

eφ) dτ (4)

To find the solvent reorganization free energies and activation free energies and to test the applicability of Marcus theory to these solutions it is sufficient to determine this function as the fluctuations in X for any given q are described by eq 5 for A(q,X), and activation free energies can be found from the points where curves of A(q,X) for different values of q cross. If Asolv(X) is a quadratic function, then each of the A(q,X) curves is parabolic and the values of X at their minima Xq are linear functions of X. These properties provide a test of Marcus theory. The existence of such a universal function is the result of the assumption that there are no changes in the non-Coulombic interactions on oxidation. Its existence obviates the need for extensive umbrella sampling. To determine Asolv(X) we consider the two partial derivatives of A(q,X)

dAsolv(X) ∂A(q,X) ∂A(q,X) ) X; )q+ ∂q ∂X dX

(7)

and note that ∂A(q,X)/∂X must be zero at points when X ) Xq, the most probable value of X for a given q. Thus

Asolv(X) ) -

∫L

q(Xq) dXq + constant

(8)

where the integral is taken along the locus of Xq. 2.2. Simulation Details. The intermolecular potentials used for the ionic liquids were taken from our earlier work.14 MeCN potentials were those of Bohm et al.23 The interactions between the solute ion and the solvent molecules were described by Coulombic terms and Lennard-Jones interactions. The values of the Lennard-Jones parameters for the intermolecular potentials between the solute ion and the solvent sites are given in Table 1. The values were based on those given by de Andrade et al.15 for the chloride-imidazolium interaction in imidazolium chlorides, although the ion diameter used (0.349 nm) is somewhat smaller than the value of 0.44 nm proposed by Smith and Dang.24 Simulations were carried out with a modified version of DL_POLY25 in a NVT ensemble, using FCC (dodecahedral) periodic boundary conditions. The distance between an atom and its nearest image was 3.4 nm for all three solutions. In addition a few simulations of [dmim][PF6] were carried out in larger cells (6.0 nm) in order to investigate system size effects which are discussed in section 3.3. The simulations with MeCN were carried out at 298 K, while those for the ionic liquids were carried out at 450 K (as these particular ionic liquids are not liquid at 298 K). A time step of 2 fs was used and the temperature was controlled by a Nose´-Hoover thermostat with a time constant of 0.5 ps. The number of molecules in the 3.4 nm simulation cells were the following: 300 MeCN + 1 solute ion; 135 [dmim][Cl] ion pairs + 1 solute ion; and 90 [dmim][PF6] ion pairs + 1 solute ion. In the 6.0 nm cell 500

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Lynden-Bell

TABLE 1: Potential Parameters for the Lennard-Jones Interactions between the Ion and Solvent Sites site

/kJ mol-1

σ/Å

Cl C1 C2 C4 N H1 H2 H4

1.009 0.664 0.664 0.7488 0.9335 0.2773 0.2773 0.2837

3.49 3.56 3.56 3.565 3.49 3.07 3.07 3.12

[dmim][PF6] ion pairs and 1 ion were used. These numbers were chosen to give pressures near 1 atm. The run lengths for each type of ion in acetonitrile were 0.9 ns while for each ion in the ionic liquids run lengths varied from 1.5 to 3ns (mostly 2 ns). The long-range Coulomb interactions are treated by the Ewald method. This method assumes that the periodic cell is uncharged, and any net charge on the sites is neutralized by a uniform background charge that exerts no forces on solute or solvent molecules. However, the influence of the periodic boundary conditions must be considered. It has been shown that if the k ) 0 (self) term in the Ewald sum is included then the periodic boundary corrections in large polar systems are small.26-29 We have chosen to exclude the k ) 0 term from the calculation of the order parameter X in the simulation and to include it in the long-range correction. In the case of a polar liquid the longrange correction (including the k ) 0 term) that has to be added to the free energy is

Alrc q )-

ζq2e2 (1 + (0)-1) 8π0L

(9)

and to the order parameter is

Xlrc ) -

2

ζqe (1 + (0)-1) 4π0L

(10)

In these equations ζ is a parameter whose value depends on the type of periodic boundary conditions used. For FCC boundaries ζ ) 3.2420, while for cubic boundaries ζ ) 2.837297. L is the distance from a point in the cell to its nearest image; 0 is the permittivity of free space and (0) is the static relative permittivity of the polar liquid. For acetonitrile, (0) ) 35.8.30 As will be discussed below, in ionic liquids the screening is complete and the term in (0)-1 should be omitted (or equivalently the static permittivity taken to be infinite). Higher order long-range corrections29 are smaller by the ratios (σ/L)2 and (σ/L)5 and are negligible for systems as large as we are using. 2.3. Analysis of Simulations. At each step, in addition to the determination of standard quantities such as the potential energy and pressure, the order parameter (which is the Coulomb energy of the solute ion divided by its charge) and the nonCoulomb interaction energy of the solute ion and the solvent were determined. Histograms of the order parameter were constructed, and the averages and rms deviations of all stored quantities were determined. In addition to site-site radial distribution functions, the spherically averaged charge density, Fq(r), was determined as a function of distance from the solute ion. This was used to find the solvent potential as a function of distance r from the solute by using

φs(r) ) φs(0) + (0)-1

∫0r Fq(s)(s2/r - s) ds

(11)

Figure 1. Plots of charges as a function of the most likely order parameter for that charge, Xq. The values of Xq are offset by +500 (MeCN) and -500 kJ/mol ([dmim][Cl]).

Figure 2. Plot of the solvent free energy, Asolv, and the solvent energy, Esolv, for [dmim][PF6]. As the curves are drawn to the same scale, their similarity shows that the entropy contribution is very small.

The first term is the potential due to the solvent at the ion, which is just equal to the order parameter divided by the charge on a proton, φs(0) ) Xq/e. 3. Results 3.1. The Solvent Free Energy, Asolv. Figure 1 shows plots of charge as a function of the most probable value of the order parameter for that charge, Xq. These data can be integrated to give the function Asolv for each of the solvents by using eq 8. The resulting plot for Asolv is shown in Figure 2 for [dmim][PF6], together with the solvent energy Esolv plotted on the same scale. The similarity of the two curves on this graph demonstrates that the variation of the solvent free energy with order parameter is primarily the result of changes in the solvent energy with very small contributions (3-4%) from changes in entropy. Similar results were found for the other ionic liquid solvents studied. 3.2. Marcus Plots. Once the solvent free energy function Asolv(X) has been determined, one can use eq 5 to find the Marcus plots for the six possible redox processes in each solvent. These are shown in Figures 3 and Figure 4 for the ionic liquid [dmim][PF6] and the polar liquid MeCN, respectively. These are the Marcus plots for the six possible redox processes. In each case the curves in red show the free energies determined directly from the probability of fluctuations in X in each simulation, while the black curves show the extended curves obtained from using Asolv and eq 5. This shows clearly that good agreement is found between the two methods but that the extended curves are necessary to obtain the activation free energies and values of the solvent reorganization free energies for the siz possible redox processes in each solvent. Figure 5

Redox Processes in Ionic Liquids

J. Phys. Chem. B, Vol. 111, No. 36, 2007 10803

Figure 3. Free energies of ions of different charges as a function of solvent order parameter, X, in the ionic liquid [dmim][PF6]. The curves in red were found from measuring probabilities while the black curves were determined from the solvent free energy, Asolv.

Figure 5. Values of the solvent reorganization free energies and the activation free energies for the six redox processes in [dmim][PF6] (above) and in the polar liquid MeCN (below).

Xq ) q[(∞)-1 - (0)-1]/(4π0s) Figure 4. Free energies of ions of different charges as a function of solvent order parameter, X, in the polar liquid MeCN. The curves in red were found from measuring probabilities while the black curves were determined from the solvent free energy, Asolv.

shows the values of these quantities in the ionic liquid [dmim][PF6] and in the polar liquid MeCN, together with estimates of the errors in the determination. If the strict form of Marcus theory were true, the free energy curves would be exact parabolae and the values of the solvent reorganization free energies would be the same for all redox processes in the same liquid. The activation free energy would be one-quarter of the solvent reorganization free energy. It can be seen from this figure that this is not true in either the ionic liquid or the polar liquid. Nevertheless, the individual free energy curves are close to being parabolae and for each redox process the difference between the three quantities λox, λred, and 4F* is comparable to the estimated uncertainties in their determination. 3.3. Screening and Local Structure. There are different possible levels of continuum models for the dielectric behavior of liquids. In the most macroscopic model of a dielectric there is a uniform field in the material with charges on interfaces.31 At this level of description an ionic liquid should be modeled as a conductor with no field within the material. A dissolved ion with charge q can be considered to be contained in a spherical cavity with a compensating charge of -q uniformly distributed over the surface. Similarly in a continuum model of a polar liquid such as MeCN, the ion potential would be partially screened by a compensating charge of -q(1 - (0)-1) or -0.97q for acetonitrile. Using such continuum model, Marcus showed2 that the order parameter, Xq, is given by

(12)

where (∞) is the permittivity at optical frequencies and s is the radius of the cavity containing the solute ion. In our model (∞) ) 1 as the molecules are not polarizable. Using this formula with the observed values of Xq, one obtains values of the cavity radius varying from 2.2 to 2.9 Å in all three solvents, depending on the ion charge. In the Debye-Hu¨ckel description of dilute solutions of electrolytes the charge of a specified ion is compensated by its ionic atmosphere, which has a finite size. This can also be described as a continuum model. Such a description is inappropriate for the high concentrations found in ionic liquids. Neither of these possible descriptions allows for the molecular nature of the solvent. Figure 6 shows the actual charge densities in the ionic liquid [dmim][PF6] and in the polar liquid MeCN as a function of distance from a dissolved ion with charge +1. It is clear that the radii estimated from eq 12 of 2.3 (MeCN) and 2.25 Å ([dmim][PF6]) are smaller than the distances of closest approach. This can be attributed to the molecular structure of the solvent near the ion. Although the positive charge on the ion induces a net negative charge in the solvent at short distances, the molecular nature of the solvent leads to charge oscillations which can be seen to extend to at least 1 nm. This means that the potential due to the ion is not completely screened by the solvent at these distances. Figure 7 shows contributions to the potential as a function of distance for the ionic liquid [dmim][PF6] and for the polar liquid MeCN. These are calculated by using eq 11 and use the measured values of Xq for q ) 1 including long-range corrections. The oscillations in the solvent potential extend to half the size of the periodic cell (17 Å). To check whether this invalidates

10804 J. Phys. Chem. B, Vol. 111, No. 36, 2007

Figure 6. Charge density in two solvents as a function of distance from an ion with charge +1. The curve for [dmim][PF6] is displaced by 0.04 e Å-3. Note that the effects of molecular structure extend to at least 1 nm.

Lynden-Bell

Figure 8. Comparison of charge densities (above) and potentials (below) in liquid [dmim][PF6] for systems with periodic cell sizes 34 (blue) and 60 Å (red).

Figure 7. Contributions to the potential as a function of distance from an ion with charge +1. The solid red curves are for the ionic liquid, [dmim][PF6], while the dotted blue curves are for the polar liquid [MeCN]. The contributions from the ion and the solvent are shown as well as the total. Note that although the ion is well screened by the solvent, the molecular structure of the solvent is apparent even at distances of 12-14 Å.

the long-range correction used, a short run for a system of one solute ion with charge q ) +1 in 500 ion pairs of [dmim][PF6] with FCC periodic boundaries and a minimum image distance of 60 Å was carried out. By using the long-range correction discussed above, the value of Xq in the larger system was determined to be 619 ( 5 kJ/mol, which can be compared with 620.4 kJ/mol in the smaller system. The difference is within computational uncertainty. Figure 8 shows a comparison of the charge density and the total potential as a function of distance from the solute ion for the small and large systems. The agreement between the large and small systems is good, although oscillations in the total potential continue toward the boundary of the larger system. We conclude that the effects of system size on the thermodynamic properties are small, even though effects on the liquid structure persist beyond 2 nm. Although we conclude that in both types of solvent investigated in this paper Marcus theory can be used and that electrochemical redox processes are dominated by long-range effects which can be described reasonably well by a continuum model, there are substantial changes in the first solvation shell of the ion. This is demonstrated in Figure 9, which shows the

Figure 9. Non-Coulombic short-range interaction energies between the solute ion and the solvent as a function of X for [dmim][PF6] (above) and MeCN (below). Note the changes in slope which correspond to changes in local coordination. However, this contribution to the solvent energy is small.

values of the non-Coulombic terms in the ion-solvent interaction (ESR IS in eq 3) as a function of the solvent order parameter X for [dmim][PF6] and MeCN. The changes in slope mark changes in the coordination number in the first shell. For example, in [dmim][PF6] the average number of anions in the first shell changes from 4 in the solutions of ions with q ) 1, to 5.5 when q ) 2, and to 6 when q ) 3. These correspond to the three branches on the left-hand side of the figure. When the solute ion is negatively charged it is surrounded by cations. The average number of cations in the first shell changes from 4 when q ) -1, to 6 when q ) -3. Although these changes affect the short-range parts of the energy, the short-range energy is less than 10% of the total solvent energy and the effect on the Marcus parameters is small.

Redox Processes in Ionic Liquids 4. Discussion and Conclusions The first conclusion that can be made from these simulations is that many of the ideas of Marcus theory are as applicable to these particular ionic liquids as to polar liquids. In both cases the free energy curves of oxidized and reduced ions are approximately parabolic, the solvent reorganization free energies are approximately equal for oxidized and reduced forms, and the activation free energies are approximately equal to onequarter of the average solvent reorganization free energy. In detail we have found that the free energy curves are not exactly parabolic so that the different redox processes considered do not have the same values for the solvent reorganization free energies and the redox activation free energies in either type of solvent. We attribute the remarkable similarity of the Marcus plots and parameters for the two ionic liquids studied here and the polar liquid MeCN to the fact that the solvent-solute interaction is due to long-range Coulomb forces. As the molecular nature of the various solvents considered affects short-range rather than long-range interactions, the continuum model of charge screening applies reasonably well. However, the effective radii of the solvent cavities are shorter than the actual cavities, reflecting the local molecular structure near the solute ions. The fact that the free energy is dominated by the energy and entropic effects are small implies that there is only a small temperature dependence of the various derived quantities such as the solvent order parameters and solvent reorganization free energies. This justifies the comparison of MeCN at 298 K with the ionic liquids at 450 K. In both types of liquid these simple redox reactions behave like classical outer sphere Marcus reactions. In a recent paper Shim and Kim19 discuss the applicability of Marcus theory to a simple electron-transfer reaction in both MeCN and [emim][PF6]. The reaction is a diatomic molecule going from an uncharged state to one with charges (e on two sites separated by 3.5 Å. The continuum theory expression for the solvent reorganization free energy in this case is

λ)

e2 ((∞)-1 - (0)-1)(1/2a1 + 1/2a2 - 1/r) (13) 4π0

where a1 and a2 are the sizes of the cavities surrounding the two charged sites and r is the separation between them. They find that the solvent reorganization free energies are 10-20 kJ mol-1 higher in the ionic liquid than in MeCN. We find that the solvent reorganization free energies for the electrochemical processes are higher in the ionic liquid [dmim][PF6] than in MeCN for negatively charged ions, but this order is reversed for positively charged ions. The actual values agree well when the 1/r term in eq 13 (which is equal to 396 kJ mol-1) is taken into account. Shim and Kim also investigated the reaction kinetics of their system and make some interesting comparisons of the short and long time behavior of solvent dynamics. We plan to do this for our system in the future, but note here that the physical processes leading to fluctuations in the solvent order parameters are quite different in the two cases (primarily reorientation in polar liquids and primarily small translational motion in ionic liquids). One concept that has been used in the discussion of the behavior of ionic liquids is that of polarity. This is a difficult concept to apply to ionic liquids as the dielectric constant (relative permittivity) is both distance- and time-dependent. It is important to consider the context in which the concept is being used. Thus in the discussion of screening around a

J. Phys. Chem. B, Vol. 111, No. 36, 2007 10805 dissolved ion, the relevant value is (0) ) ∞, although in other contexts the polarity is similar to that of alcohols. There have been a number of investigations of the electrostatic screening in ionic liquids, mostly of the constituent ions. Del Po´polo and Voth16 showed that the screening length in [emim][NO3] is about 7 Å, and Kim and co-workers18 found values of about 9-10 Å in [emim][Cl] and [emim][PF6]. Kobrak20 investigated the electrostatic screening around a dye molecule in an ionic liquid and Pinilla et al. looked at the screening near a charged wall.32 In all these cases the length scales were comparable to those we observe here. We note that if quantitative comparisons between this type of calculation and either experimental or ab initio calculations are to be made, it is important to include a correction for the difference in the optical permittivities. Finally one can ask the question as to whether Marcus theory applies to all ionic liquids. All we have shown here is that it applies to two examples with unusually small cations. Similar conclusions were reached by Shim and Kim19 for charge-transfer reactions in ethylmethylimidazolium hexafluorophsophate. These are important demonstrations as Marcus theory was developed for polar liquids and not for ionic liquids. Although the arguments given earlier about the importance of long-range electrostatic forces suggest that Marcus theory could apply generally to ionic liquids, the increased heterogeneity and slower dynamics found in liquids with larger cations suggest that the redox process in some ionic liquids may be more complex. For example, there is the possibility of oxidation to an ion in a metastable solvation state followed by a very slow relaxation to a stable state. Acknowledgment. I thank Richard Compton for stimulating this work and discussing the results, and Michiel Sprik, Joachim Blumberger, and Chris Hardacre for useful discussions. References and Notes (1) Silvester, D. S.; Compton, R. G. Z. Phys. Chem. 2006, 220, 1247. (2) Marcus, R. Annu. ReV. Phys. Chem. 1964, 15, 155. (3) Nitzan, A. Chemical Dynamics in Condensed Phases; Oxford University Press: Oxford, 2006. (4) Warshel, A. J. Phys. Chem. 1982, 86, 2218. (5) King, G.; Warshel, A. J. Chem. Phys. 1990, 93, 8682. (6) Kuharski, R. A.; Bader, J. S.; Chandler, D.; Sprik, M.; Klein, M. L.; Impey, R. W. J. Chem. Phys. 1988, 89, 3248. (7) Hartnig, C.; Koper, M. T. M. J. Chem. Phys. 2001, 115, 8540. (8) Yelle, R. B.; Ichiye, T. J. Phys. Chem. 1997, 101, 4127. (9) Dubois, V.; Archirel, P.; Boutin, A. J. Phys. Chem. B 2001, 105, 9363. (10) Blumberger, J.; Sprik, M. Lecture notes for the International School of Solid State Physics-34th course: Computer Simulations in Condensed Matter: from Materials to Chemical Biology, Erice, Sicily, July 20-August 1, 2005. Lect. Notes Phys. 2006, 704, 467. (11) Blumberger, J.; Bernasconi, L.; Tavernelli, I.; Vuilleumier, R.; Sprik, M. J. Am. Chem. Soc. 2004, 126, 3928. (12) VandeVondele, J.; Lynden-Bell, R. M.; Meijer, E. J.; Sprik, M. J. Phys. Chem. B 2006, 110, 3614. (13) Blumberger, J.; Sprik, M. Theor. Chem. Acc. 2006, 115, 113. (14) Hanke, C. G.; Price, S. L.; Lynden-Bell, R. M. Mol. Phys. 2001, 99, 801. (15) de Andrade, J.; Boes, E. S.; Stassen, H. J. Phys. Chem. B 2002, 106, 13344. (16) Del Po´polo, M. G.; Voth, G. A. J. Phys. Chem. B 2004, 108, 1744. (17) Shim Y; Choi M. Y.; Kim H. J. J. Chem. Phys. 2005, 122, 044511. (18) Shim, Y.; Choi, M. Y.; Kim, H. J. J. Chem. Phys. 2005, 122, 044510. (19) Shim, Y.; Kim, H. J. J. Phys. Chem. B 2007, 111, 4510. (20) Znamenskiy, V.; Kobrak, M. N. J. Phys. Chem. B 2004, 108, 1072. (21) Kobrak, M. N. J. Chem. Phys. 2006, 125, 064502. (22) Lynden-Bell, R. M. Electrochem. Commun. 2007, 9, 1857. (23) Bohm, H. J.; McDonald, I. R.; Madden, P. A. Mol. Phys. 1983, 49, 347. (24) Smith, D. E.; Dang, L. X. J. Chem. Phys. 1994, 100, 3757.

10806 J. Phys. Chem. B, Vol. 111, No. 36, 2007 (25) Smith, W; Forester, T. R. The DL_POLY manual, Daresbury Laboratory, 1996; http://www.cse.scitech.ac.uk/ccg/software/DL_POLY/ index.shtml. (26) Hummer, G.; Pratt, L. R.; Garcı´a, A. E. J. Chem. Phys. 107, 9275. (27) Hummer, G.; Pratt, L. R.; Garcı´a, A. E.; Neumann, M. In Electrostatic Interactions; Pratt, L. R., Hummer, G., Eds.; AIP Conference Proceedings 492, American Institute of Physics, 1999. (28) Lynden-Bell, R. M.; Rasaiah, J. C. J. Chem. Phys. 1997, 107, 1981.

Lynden-Bell (29) Hu¨nenberger, P. H. In Electrostatic Interactions; Pratt, L. R., Hummer, G., Eds.; AIP Conference Proceedings 492; American Institute of Physics, 1999. (30) Barthel, J.; Kleebauer, M.; Buchner, R. J. Solution Chem. 1995, 24, 1. (31) Allen, R. J.; Hansen, J. P.; Melchionna, S. Phys. Chem. Chem. Phys. 2001, 3, 4177. (32) Pinilla, C.; Del Po´polo, M. G.; Kohanoff, J.; Lynden-Bell, R. M. J. Phys. Chem. B 2007, 111, 4877.